Finance and Stochastics

, Volume 20, Issue 1, pp 219–265 | Cite as

Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility

  • José E. Figueroa-López
  • Sveinn Ólafsson


In Figueroa-López et al. (Math. Finance, 2013), a second order approximation for at-the-money option prices is derived for a large class of exponential Lévy models, with or without a Brownian component. The purpose of the present article is twofold. First, we relax the regularity conditions imposed on the Lévy density to the weakest possible conditions for such an expansion to be well defined. Second, we show that the formulas extend both to the case of “close-to-the-money” strikes and to the case where the continuous Brownian component is replaced by an independent stochastic volatility process with leverage.


Exponential Lévy models Stochastic volatility models Short-time asymptotics ATM option pricing Implied volatility 

Mathematics Subject Classification (2010)

60G51 60F99 91G20 

JEL Classification




The authors gratefully acknowledge the constructive and helpful comments provided by two anonymous referees, which significantly contributed to improve the quality of the manuscript.


  1. 1.
    Abundo, M.: Some remarks on the maximum of a one-dimensional diffusion process. Probab. Math. Stat. 28, 107–120 (2008) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersen, L., Lipton, A.: Asymptotics for exponential Lévy processes and their volatility smile: survey and new results. Int. J. Theor. Appl. Finance 16, 1–98 (2013) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carr, P., Madan, D.: Saddle point methods for option pricing. J. Comput. Finance 13, 49–61 (2009) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall, London (2004) zbMATHGoogle Scholar
  5. 5.
    Figueroa-López, J.E., Gong, R., Houdré, C.: High-order short-time expansions for ATM option prices of exponential Lévy models. Math. Finance (2014). doi: 10.1111/mafi.12064 Google Scholar
  6. 6.
    Figueroa-López, J.E., Forde, M.: The small-maturity smile for exponential Lévy models. SIAM J. Financ. Math. 3, 33–65 (2012) zbMATHCrossRefGoogle Scholar
  7. 7.
    Figueroa-López, J.E., Houdré, C.: Small-time expansions for the transition distributions of Lévy processes. Stoch. Process. Appl. 119, 3862–3889 (2009) zbMATHCrossRefGoogle Scholar
  8. 8.
    Gao, K., Lee, R.: Asymptotics of implied volatility to arbitrary order. Finance Stoch. 8, 349–392 (2014) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Houdré, C.: Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30, 1223–1237 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kallenberg, O.: Foundations of Modern Probability, 1st edn. Springer, Berlin (1997) zbMATHGoogle Scholar
  11. 11.
    Mijatović, A., Tankov, P.: A new look at short-term implied volatility in asset price models with jumps. Math. Finance (2013). doi: 10.1111/mafi.12055 Google Scholar
  12. 12.
    Muhle-Karbe, J., Nutz, M.: Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab. 48, 1003–1020 (2011) zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Roper, M.: Implied volatility: general properties and asymptotics. Ph.D. Thesis, University of New South Wales (2009). Available at
  14. 14.
    Rosenbaum, M., Tankov, P.: Asymptotic results for time-changed Lévy processes sampled at hitting times. Stoch. Process. Appl. 121, 1607–1633 (2011) zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rüschendorf, L., Woerner, J.: Expansion of transition distributions of Lévy processes in small time. Bernoulli 8, 81–96 (2002) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  17. 17.
    Schoutens, W.: Lévy Processes in Finance. Wiley, West Sussex (2003) CrossRefGoogle Scholar
  18. 18.
    Tankov, P.: Pricing and hedging in exponential Lévy models: review of recent results. In: Carmona, R.A., et al. (eds.) Paris–Princeton Lecture Notes in Mathematical Finance. Lectures Notes in Mathematics, vol. 2003, pp. 319–359. Springer, Berlin (2010) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA
  2. 2.Department of StatisticsPurdue UniversityWest LafayetteUSA

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